Lie - Poisson Deformation of the Poincaré Algebra

We find a one parameter family of quadratic Poisson structures on R 4 × SL(2, C) which satisfies the property a) that it is preserved under the Lie-Poisson action of the Lorentz group, as well as b) that it reduces to the standard Poincaré algebra for a particular limiting value of the parameter. (The Lie-Poisson transformations reduce to canonical ones in that limit, which we therefore refer to as the 'canonical limit'.) Like with the Poincaré algebra, our deformed Poincaré algebra has two Casimir functions which we associate with 'mass' and 'spin'. We parametrize the symplectic leaves of R 4 × SL(2, C) with space-time coordinates, momenta and spin, thereby obtaining realizations of the deformed algebra for the cases of a spinless and a spinning particle. The formalism can be applied for finding a one parameter family of canonically inequivalent descriptions of the photon.